---
tag: B7948M365K
kind: concept
name: Regular boundary point
msc: [31B25, 31B20, 31A25]
sources:
  - url: https://en.wikipedia.org/wiki/Perron_method
---

## Summary

A boundary point of a bounded open set is regular if the generalized solution of the [[Dirichlet problem|8D56GQ4AG8]] tends to the prescribed value there.
This must hold for every continuous boundary function.
The Dirichlet problem is solvable for all continuous data exactly when every boundary point is regular.
Regularity is a local property: it holds at $\xi$ if and only if there is a [[barrier|ED0SZS1VED]] at $\xi$.
It is characterized by Wiener's criterion in terms of capacity, and the irregular points always form a polar set.

## Explanations

### L1 human

At most boundary points the solution of the Dirichlet problem behaves well and takes the boundary value you prescribe.
Some points are so small or so thin that the solution does not notice them.
The centre of a punctured disc is the simplest such irregular point: whatever value you prescribe there, the solution ignores it.
A point on a smooth piece of boundary, or the corner of a square, is regular.